Let there be 2 such objects, whose coterminus edges are identical. Then, the volume of the tetrahedron is th of that of the parallelopiped. The expression can also be obtained using the following theorem :

We were introduced to the trigonometric functions in class with their general definitions for any angle . The specialty of these functions is that their values get repeated after an interval. This is because after every interval of , the angles have same initial and terminal arms.

Basics : If we fix the initial arm of the angle as positive axis, with vertex at origin. Let be ANY point on the terminal arm of the angle. Let be . Then,

Periodicity : The characteristic of trigonometric functions, which repeat their values after a fixed interval, is known as periodicity. The smallest non-negative interval is known as the fundamental period.

Trigonometric Ratios of Standard Angles

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Equations and Solutions

The value which satisfies an equation is known as a solution. For example, is an equation. On simplifying,

So, the solution is .

Trigonometric Equations

The equations involving trigonometric functions are known as trigonometric equations. The unknown values, which are to be found, represent the measure of an angle. For example, is satisfied by or .

Principal Solutions

The solution, which lies between and , is known as the principal solution.

General Solutions

We saw that the measure of full circle is radians or . Hence, after adding to above angle, we will still get a solution. So,

Similarly,

Thus, there are infinitely many solutions to above equations, apart from . These solutions are known as general solutions. Consider another example:

Note: We will have to use allied angles formulas and the formula sheet to solve problems of this kind, where we are asked to find the general solutions.

Polar Coordinates

The Cartesian coordinates and the polar coordinates are inter-convertible. See the figure below:

Solving a Triangle

A triangle has 3 sides, say, and 3 angles, . By solving a triangle, we mean, to find values of all lengths of sides and all angles of a triangle. See the figure below.

The side opposite to is denoted by and so on.

There are certain rules to be used to solve these problems. These are:

Sine Rule

is the radius of the circumcircle.

Cosine Rule

Projection Rule

Half Angle Formulas

is the semi-perimeter.

Area of a Triangle

The area of triangle is given by,

It is also given by

. This is known as Heron’s formula.

Napier’s Analogies

Inverse Trigonometric Functions

If a function is defined from set A to set B (which is one-one and onto), we can define an inverse function, from set B to set A. So, if , then and .

If , then . If , then .

Note that is different from . is , which is .

On the other hand, is the angle, whose sine is .

Corresponding to each of six trigonometric functions, we have 6 inverse trigonometric functions, i.e. .

Recall : Principal value is the value of angle, which lies between and .

Standard Limits

Introduction

Consider 2 functions, and . Let be a value at which these functions are defined.

I) If and , then the limit takes the form .

II) If , then takes the form

III) If and , then the limit takes the form

IV) If , and , then the limit takes the form .

V) If , then the limit takes the form .

VI) If and and , then the limit takes the form

VII) If and and , then the limit takes the form .

These are known as the indeterminate forms. The limits are evaluated either by L’Hosptial’s rule or by substituting an equivalent infinitesimal.

L’Hosptial’s Rule (French : )

The rule can be proved using Taylor’s theorem. It says, if and are at or and , then

This rule is sometimes applied on th derivatives, if all derivatives of lesser orders are .

Equivalent Infinitesimal

This is used for evaluation of form. One of the functions can be replaced by another, if they both converge to at a point and the limit of their ratio at that point is . For example,

Hint for MCQs

One can try substituting a value of closer to (but not equal to) the actual limit. Evaluate the function using the calculator. The answer will be closer to the actual limit. We’ve actually used the concept of limit here.

Explanation: Consider the limit

This is of the form . Let’s put in the function .

We get as .

On substituting , a value closer to , we get .

On substituting , a value closer to and , we get . Clearly, the limit is .